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\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex

\input BoxedEPS
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\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue -10pt



\cl {\bf About  the Classical Enneper Surface}
\cl {\bf and some Polynomial Relatives }
\lf
\cl { See also: About Minimal Surfaces}
\lf
\cl{\sc  Definition with explicit formulas}
\LF
The {\it classical Enneper Surface} is a minimal immersion of the complex plane, $\BC$,
into Euclidean space $\BR^3$. It is given by the formula

\cl{$F(z) :=  \hbox{real}(\{  z^3/3 -z,\  i*(z^3/3 + z),\  z^2 \})  $.} 
\noindent
If one wants to see coordinate lines on the image one can use Cartesian coordinates
for the complex plane, $z := x + i\cdot y$, or polar coordinates 
$z := r\cdot(\cos\varphi  + i\cdot\sin\varphi )$, and map those grids with $F$. The Cartesian choice is natural here since 
its parameter lines are principal curvature lines. However the polar choice also has  merits---namely, rotations around the origin are isometries and the coordinate 
lines are orbits. Moreover all symmetry lines of the surface are radial parameter lines.
The Action Menu of 3DXM allows to switch between these parametrizations.
\lf
In 3DXM one can also deform this classical surface, but we need to explain the significance of
what one sees. See the last page of this text.
\LF
Early in the second half of the nineteenth century the {\it Enneper-Weierstra\ss\ representation}
of minimal surfaces was discovered. Its main advantage is that it permits one to write a formula
for a minimal surface in terms of important geometric quantities. Every surface in $\BR^3$
can be mapped to the 2-sphere $\BS^2$ by sending each point on the surface to the unit
normal at this point; this map is called the geometric Gau\ss\ map $N$. For minimal surfaces this
map is angle preserving, but orientation reversing. Composition of $N$ with the orientation
reversing stereographic projection therefore gives a map $g$ from the surface into $\BC$ which is both 
{\it  and\/} orientation preserving. Finally, if we interpret $90^\circ$ rotation on each tangent
space of a surface in $\BR^3$ as multiplication of tangent vectors by $i$, then with this convention
$g$ becomes a meromorphic function, the meromorphic Gau\ss\ map of the minimal surface.
This meromorphic Gau\ss\ map is one-half of the {\it Weierstra\ss\ data} which are needed to
write down the { Enneper-Weierstra\ss\ representation}. The remaining part of these data is
the differential $dF^3$ of the third component of $F$, i.e., of the height function on the minimal
surface. It might seem at first that we must know a minimal surface rather well before we
have its Weierstra\ss\ data. However, on a large class of geo\-metrically important minimal
surfaces the situation is simple indeed. If a minimal surface is complete and has finite total
curvature then the Gau\ss\ map $g$ is determined---up to a constant factor---by its zeros
and poles, in other words by its vertical normals. This important result extends to differentials,
in particular to the differential of the height function, after we perform a small trick, namely
extend the real valued differential $dF^3$ to a complex valued one by putting for every
tangent vector $v$ of the surface
$$\eqalign{
dh(v)&:= dF^3(v) -i\cdot dF^3(i\cdot v)   \cr
         &:= dF^3(v) -i\cdot dF^3(\hbox{Rot}^{90}( v)) .
}$$
To make matters even simpler, observe that the points on the surface, where the normal is
vertical, are the same points where the differential of the height function is zero. More precisely,
the zeros and poles  of  $g$ on the minimal surface are precisely the zeros of $dh$, even with the same multiplicity. (To complete this discussion we would have to study the situation at infinity, but we will omit this.)
The main point is to point out that very few, {\it finitely many}, data about such 
minimal surfaces suffice to find their Weierstra\ss\ data and therefore explicitly parametrize them.
Here is this famous formula:  %\goodbreak

\centerline {Weierstra\ss\ Representation in terms of {\it g, dh}:}  
\vskip -15pt
$$ \eqalign{
 F(&z) := \hbox{Re}\left(
\int_*^z \left\{{1\over 2}\left(g - {1\over g}\right)\,dh,{i\over 2}\left(g + {1\over g}\right)\,dh,\ dh\right\}\right)
}$$
The classical Enneper surface is obtained if we put $g(z)=z,\ dh = zdz$.
\lf
This generalizes to polynomials $P(z)$, put:\lf
$g(z)=P(z),\ dh = P(z)dz$.  \Lf
\hrule\noindent
3DXM allows $P(z):= aa\cdot z+bb\cdot z^2 +cc\cdot z^3$. \Lf
\hrule\noindent
The pure powers have again the rotations around the origin as metric isometry group and
{\it polar coordinates} provide a much better view of these surfaces and are recommended. 
All surfaces of the associate family are, for $g(z)=z^k$, congruent. There
are straight lines on the surface, and if one looks in the direction of the z-axis onto the surface,
then the portion below these lines is drawn first. The default morph deforms two such surfaces
into each other.

\Lf
\bye
\lf\phantom{.}\hskip1cm





